C1 C1
Uw
1\
:,1
\
I,
95
Madow. ~/
The conditions are usually satisfied with regard to estimates
from sample surveys.
As a rule of thumb the variance formula is usually
accepted
'as satisfactory if the coefficient of variation of the variable
o
w
in the denominator is less than 0.1; that is, if -- < 0.1.
In other words,
w
this condition states that the coefficient of variation of the estimate in
the denominator should be less than 10 percent.
A larger coefficient of
variation might be tolerable before becoming concerned about Equation (3.26)
as an approximation.
o
The condition ~
< 0.1 is more strin~ent than necessary for reg~rding
w
the bias of a ratio as negligible.
With few exceptions in practice the
bias of a ratio is ignored.
Some of the logic for this ,,,illappear in
the illustration below.
To summarize, the conditions when Equations (3.25)
and (3.26) are not gpod approximations are such that the ratio is likely to
be of questionable value owing to large variance.
If u and ware
linear combinations of random variables, the theory
presented in previous sections applies to u and to w. Assumin~ u and w
are estimates from a sample,
u take into
the
to estimate Var(-)
account
w
sample design and substitute in Equation (3.26) estimates of u, W,
2
2
0u' o ,
w
and p
• Ignore Equation (3.25) unless there is reason to believe the bias
uw
of the ratio mi~ht be important relative to its standard error.
It is of interest to note the similairity between Var(u-w) and var<;).
According to Theorem 3.5,
Var(u-w) g 02 + 02 - 2p
u
w
uw
JJ Hansen, Hurwitz, and Hadow, Sample Survey Methods and Theory,
Volume I, Chapter 4, John Wiley and Sons, 1953.
96
By definition the relative variance of ~n estimate is the variance of the
estimate divided by the s~uare of its expected value.
Thus, in terms of
7
the relativ~ variance of a ratio, Equation (3.26) can be written
2
2
a
a
ReI Var(~) ••~ + ~ - 2p
w
-2
-2
uw
u
w
a au ,,,
uw
The similarity is an aid to remembering the formula for Var(u).
w
elements from a population of N. Let x and y be t